Let Fq[T] be the polynomial ring over a finite field Fq. We study endomorphism rings of Drinfeld Fq[T]-modules arbitrary rank fields. compare to their subrings generated by Frobenius and deduce from this refinement reciprocity law for division fields modules proved in our earlier paper. then use these results give an efficient algorithm computing discuss some interesting examples produced algor...

Let D be a division ring, T be a variable over D, σ be an endomorphism of D, δ be a σ-derivation on D and R = D[T ;σ, δ] the left skew polynomial ring over D. We show that the set (V alν(R), ) of σ-compatible real valuations which extend to R a fixed proper real valuation ν on D has a natural structure of parameterized complete non-metric tree, where is the partial order given by μ μ̃ if and onl...

-We describe three algorithms to count the number of points on an elliptic curve over a finite field. The first one is very practical when the finite field is not too large; it is based on Shanks’s baby-step-giant-step strategy. The second algorithm is very efficient when the endomorphism ring of the curve is known. It exploits the natural lattice structure of this ring. The third algorithm is ...

Let R be a Cohen-Macaulay ring and M a maximal CohenMacaulay R-module. Inspired by recent striking work by Iyama, BurbanIyama-Keller-Reiten and Van den Bergh we study the question of when the endomorphism ring of M has finite global dimension via certain conditions about vanishing of Ext modules. We are able to strengthen certain results by Iyama on connections between a higher dimension versio...

Let R be an associative ring. A map σ : R → R is called a ring endomorphism if σ(x+y) = σ(x)+σ(y) and σ(xy) = σ(x)σ(y) for all elements a,b ∈ R. A ring R is said to be rigid if it has only the trivial ring endomorphisms, that is, identity idR and zero 0R . Rigid left Artinian rings were described by Maxson [9] and McLean [11]. Friger [4, 6] has constructed an example of a noncommutative rigid r...

let $gamma$ be a normal subgroup of the full automorphism group $aut(g)$ of a group $g$‎, ‎and assume that $inn(g)leq gamma$‎. ‎an endomorphism $sigma$ of $g$ is said to be {it $gamma$-central} if $sigma$ induces the the identity on the factor group $g/c_g(gamma)$‎. ‎clearly‎, ‎if $gamma=inn(g)$‎, ‎then a $gamma$-central endomorphism is a {it central} endomorphism‎. ‎in this article the conditi...

Let G be an abelian p-group sum of finite homocyclic groups Gi. Here, we determine in which cases the automorphism group of G splits over kerσ, where σ :Aut(G) → ∏ iAut(Gi/pGi) is the natural epimorphism. 1. Preliminaries Throghout this paper, p is an arbitrary prime and r is a fixed ordinal number. Let G = ⊕i≤rGi be an abelian p-group, such that Gi is an homocyclic group of exponent pi and fin...

We show that for a large class of rings R, the number principally polarized abelian varieties over finite field in given simple ordinary isogeny and with endomorphism ring R is equal either to 0, or ratio numbers associated up some small computable factors. This includes maximal order CM K (for which result was already known), as well generated Z by Frobenius Verschiebung.